I don’t have Gelman’s book, but: logistic regression says p = 1 / (1 + exp(-z)) where z is a linear combination of 1 and the independent variables. But then z is just the “log odds”, log(p/(1-p)); you can think of the coefficient of 1 as being the log prior odds ratio and the other coefficients as being the amount of evidence you get for X over not-X per unit change in each independent variable.
I don’t have Gelman’s book, but: logistic regression says p = 1 / (1 + exp(-z)) where z is a linear combination of 1 and the independent variables. But then z is just the “log odds”, log(p/(1-p)); you can think of the coefficient of 1 as being the log prior odds ratio and the other coefficients as being the amount of evidence you get for X over not-X per unit change in each independent variable.